Why is my texture appearing differently in texture mode compared to render display mode?
My guess would be that you haven't saved the image after texture painting and/or reloaded it so Cycles isn't seeing the updated texture yet.

If you are in texture paint mode you can choose 'Save All Images' from the 'Slots' tab of the toolbar to save any edited images to disk:



You can then reload the image node in the node editor by selecting the image node and from the 'Properties' panel (Press N while in the node editor) and choose the 'Reload' button:



Note: I have sometimes found the need to switch from rendered view to another view and then back to rendered view for the image texture to update.